When my DW and I took our first cruise back in 2003, we were worried about that same thing. So, we researched seasickness extensively and stocked up on all kinds of remedies. Little wrist straps with a button thing on them, pills, etc.

Decided to just go with the pills and hope for the best. So, we started taking the Dramamine Less Drowsy formula the night before the cruise. Then one every night for the duration of the cruise.

We've followed this regimen on every cruise we've been on since and you know what? We've never had even a hint of a problem. Not a bit. Whatever it is that we've done, it's worked and worked well.

Now, the other side of this whole argument is that the ships are huge! The only time I've ever experienced anything like motion on a Disney Cruise was on the first night of our last cruise. According to what our own Slowhand has told me, he was informed that this was due to us crossing the Gulf Stream that night. But even at that, it wasn't bad. It was more like a gentle swaying than anything else. And by the next morning it was completely gone and the ship's motion was virtually undetectable.

There are two other places that I've noticed the motion of the ship.

The first is lying in bed at night before going to sleep. But believe me, I want to put that experience in a can and bring it back home with me. It's a gentle, humming, rocking sensation. Not rough. Just a very gentle back and forth, up and down vibration kind of feeling. Wonderfully pleasant and guaranteed to produce the soundest sleep you've had in years.

The second is oddly enough, in the evening while having an adult beverage in Sessions. In Sessions on the Disney Magic, they have these very large, almost floor to ceiling portholes. Now, Sessions is located on deck 3 which is above the water line but still low enough that you can see the water out these portholes. At night, the outside of the ship is lighted a bit and the light from inside spills out so that the water outside these portholes is kind of illuminated. As the ship is making speed, the water rushes by under these and the sensation of motion is dramatically enhanced. Looking out these portholes, the view was making me a bit queasy. The simple answer to that one is "don't do that". Do what I did. Have another of your favorite adult beverage and enjoy the pleasant company that you're with. Listen to the piano player and forget the fact that these portholes are even there. Do that, and you won't have a problem.

Seriously, in our three (soon to be four) Disney cruises, I've never heard of anyone complaining of sea sickness. And if anyone were going to have a problem with it, my DW would be the one!

Donna and her BFF Bev were both concerned about this when we went on our cruise in 2005. It was a bit rough to them as we headed out to sea but the Capt. told them during an introductory meeting (Bev's DH and I were in the bar watching a football game) that once we were at sea all would be OK and that was indeed the truth. Neither of them took anything the whole trip. The Disney ships have great stabilizers.

Tom has indeed explained the situation in a nutshell.
Just to add, the DCL ships [along with probably ALL similar vessels] are equipped with stabalisers which the Captain deploys in times of need. These are 25 feet long "wings" that protrude from below the waterline on the hull which alleviates the swaying motion of the ship and they trim themselves to suit the conditions.

We, along with many others however were a bit queasy on the first night of our first DCL cruise. As it was our first cruise I didn`t know if this was normal but the boat was sure rocking that evening....at least I managed to walk in a straight line to my stateroom after a few bedtime drinks in Sessions, rather than the customary swagger!
We found out the next day that this had in fact been the worst night ever that the ship and it`s crew had experienced....just our luck!

We have since done a further 8 DCL cruises and we have never had a problem. In a funny sort of way when I get back on dry land I rather miss the gentle movement of the boat which sort of rocks me to sleep!